Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells

Abstract Recent experiments by Pihler-Puzovic et al. (Phys. Rev. Lett., vol. 108, 2012, article 074502) have shown that the onset of viscous fingering in circular Hele-Shaw cells in which an air bubble displaces a viscous fluid is delayed considerably when the top boundary of the cell is replaced by an elastic membrane. Non-axisymmetric instabilities are only observed at much larger flow rates, and the large-amplitude fingers that develop are fundamentally different from the highly branched fingers in rigid-walled cells. We explain the mechanism for the suppression of the instability using a combination of linear stability analysis and direct numerical simulations, based on a theoretical model that couples a depth-averaged lubrication equation for the fluid flow to the Föppl–von Kármán equations, which describe the deformation of the elastic membrane. We show that fluid–structure interaction affects the instability primarily via two changes to the axisymmetric base flow: the axisymmetric inflation of the membrane prior to the onset of any instabilities slows down the expansion of the air bubble and forces the air–liquid interface to propagate into a converging fluid-filled gap. Both of these changes reduce the destabilizing viscous effects that drive the fingering instability in a rigid-walled cell. In contrast, capillary effects only play a very minor role in the suppression of the instability.